Distorted exported .fbx model when objects are under parent
I'm not sure if this problem has been resolved or not, but for future reference, I also frequently work with Unity and when FBX exports are distorting for me, it usually involves one of two things:

1) As mentioned in comments, I have a mesh that has an unapplied transform to it, and the FBX exporter is getting confused.

2) I have an animation doing something wonky on its first keyframe, which needs to be replaced or removed. This can result in distortions which look nothing like the animation itself, particularly if there was unapplied scaling or rotation involved at some point.

This may or may not solve the issue, but I'd bet that it's an animation. Perhaps the model has an entry for the leaves rustling or blowing with the wind?